This is a brief review of the progress made since the seminal contributions to the foundations of the theory of nucleosynthesis by M. Burbidge, G. Burbidge, Fowler and Hoyle, and by Cameron. The reviewed topics are (1) the nuclear physics input to the nucleosynthesis models (nuclear masses, fission, rates of β-decays, neutrino reactions, photoreactions, and nuclear charged particle-induced or neutron-induced reactions), (2) the nuclear physics and astrophysics aspects of the r-process, and (3) the same items for the p-process.

Primordial nucleosynthesis is one of the three observational evidences for the Big–Bang model. Even though they span a range of nine orders of magnitude, there is indeed a good overall agreement between primordial abundances of 4He, D, 3He and 7Li either deduced from observation or primordial nucleosynthesis. This comparison was used for the determination of the baryonic density of the Universe. For this purpose, it is now superseded by the analysis of the CMB anisotropies. Big–Bang nucleosynthesis remains, nevertheless, a valuable tool to probe the physics of the early Universe.

In this paper, I explore various transport processes that have a large impact of the distribution of elements inside stars and thus, on stellar evolution. A heuristic description of the physics behind equations is provided, and key references are given. Finally, for each process, I will briefly review (some) important results as well as discuss directions for future work.

We review the structural and nucleosynthetic evolution of stars in the mass range 0.8–11 M๏, focusing on the AGB and super-AGB phases. The effects of transport processes associated with rotation and overshooting are described and their impact on nucleosynthesis analyzed. We present selected observational properties of AGB stars and discuss the weaknesses and failures of current stellar modeling.

We review the information that planetary nebulae and their immediate progenitors, the post-AGB objects, can provide to probe the nucleosynthesis and mixing in low- and intermediate-mass stars. We emphasize new approaches based on high signal-to-noise spectroscopy of planetary nebulae and of their central stars. We mention some of the problems still to overcome. We emphasize that, as found by several authors, planetary nebulae in low-metallicity environments cannot be used to probe the oxygen abundance in the interstellar medium out of which their progenitors were formed, because of abundance modification during stellar evolution.

We review some important observed properties of massive stars. Then we discuss how mass loss and rotation affect their evolution and help in giving better fits to observational constraints. Consequences for nucleosynthesis at different metallicities are discussed. 
Mass loss appears to be the key feature at high metallicity, while rotation is likely dominant at low and very low metallicities. We discuss various indications supporting the view that very metal-poor stars had their evolution strongly affected by rotational mixing. Many features, like the origin of primary nitrogen at low metallicity, that of the C-rich extremely metal-poor halo stars, of He-rich stars in massive globular clusters, of the O-Na anticorrelation in globular clusters may be related to the existence of a population of very fast rotating metal-poor stars that we tentatively call the spinstars. A fraction of these spinstars may also be the progenitors of GRB in very metal-poor regions. They may avoid pair instability explosion due to the heavy mass loss undergone during their early evolutionary phases and be, dependent on their frequency, important sources of ionising photons in the early Universe.

We review the presupernova evolution, the explosion and the explosive nucleosynthesis of massive stars in the range 11–120 M๏. We mainly focus on the advanced evolutionary phases prior to the explosion, i.e. from the core He exhaustion up to the onset of the iron core collapse.

Type Ia supernovae are thought to be the outcome of the thermonuclear explosion of a carbon–oxygen white dwarf in a close binary system. In this review we discuss the observational background, the possible progenitors, the mechanisms leading to the instability and the development of the explosion itself as well as the remaining uncertainties.

The formalism of the simple model of galactic chemical evolution (GCE) and its main ingredients (stellar properties, initial mass function, star formation rate and gas flows) are presented in this tutorial. It is stressed that GCE is not (yet) an astrophysical theory, but it merely provides a framework in which the large body of data concerning the chemical composition of stars and gas in galaxies may be intrepreted. A few examples illustrating those concepts are provided, through studies of the Solar neighborhood and of the Milky Way's halo.

The detailed abundance patterns in extremely metal-poor (EMP) stars are examined, using the results of the First Stars ESO-VLT Large Program on large samples of dwarfs (located close to the main sequence turnoff) and giant stars (Cayrel et al. 2004; Bonifacio et al. 2007 and subsequent papers). Abundances measured in giants are compared to those measured in turnoff stars, and to comparable quality data in the literature, to demonstrate that the systematics involved in most abundance ratios are very small. A few exceptions are discussed: a reanalysis of Na and Al in EMP samples is performed using NLTE computations. In addition, giants with luminosities brighter than the RGB bump are shown to exhibit the sign of surface pollution by mixing of the products of H-burning, affecting the measured abundances of C, N and Li. 
The implications of a few key abundance ratios (among those considered to be robust and representative of the early galactic gas) are examined in terms of the early Milky-Way enrichment processes and the possible nucleosynthesis contributors (PISNs, SNII, hypernovae). The very low dispersion found around the mean trends is also discussed in terms of chemical enrichment processes.

Galactic globular cluster (GC) stars exhibit abundance patterns which are not shared by their field counterparts, namely the well-documented C-N, O-Na and Mg-Al anticorrelations. Recent observations provided compelling evidence that these abundance anomalies were already present in the intracluster gas from which the presently observed stars formed. A widely held hypothesis is that the gas was polluted early in the history of the GC by material processed through H-burning at high temperature and then lost by stars more massive than the observed long-lived stars. However the “polluters" have not been unambiguously identified yet. Most studies have focused on AGB stars, but rotating massive stars present an interesting alternative. Here we critically analyse the pros and cons of both potential stellar polluters. We discuss the constraints that the observational data bring on stellar nucleosynthesis and hydrodynamics as well as on nuclear reaction rates and we try to answer to the following question: “Are GC ideal laboratories to test stellar nucleosynthesis and hydrodynamics?"